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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochocsn | Structured version Visualization version GIF version |
Description: The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015.) |
Ref | Expression |
---|---|
dochocsn.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochocsn.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochocsn.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochocsn.v | ⊢ 𝑉 = (Base‘𝑈) |
dochocsn.n | ⊢ 𝑁 = (LSpan‘𝑈) |
dochocsn.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochocsn.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
dochocsn | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝑋})) = (𝑁‘{𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochocsn.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dochocsn.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | dochocsn.o | . . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
4 | dochocsn.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
5 | dochocsn.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
6 | dochocsn.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | dochocsn.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
8 | 7 | snssd 4808 | . . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
9 | 1, 2, 3, 4, 5, 6, 8 | dochocsp 40789 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘{𝑋})) |
10 | 9 | fveq2d 6895 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋}))) = ( ⊥ ‘( ⊥ ‘{𝑋}))) |
11 | eqid 2727 | . . . . 5 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
12 | 1, 2, 4, 5, 11 | dihlsprn 40741 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
13 | 6, 7, 12 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
14 | 1, 11, 3 | dochoc 40777 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋})) |
15 | 6, 13, 14 | syl2anc 583 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋})) |
16 | 10, 15 | eqtr3d 2769 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝑋})) = (𝑁‘{𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {csn 4624 ran crn 5673 ‘cfv 6542 Basecbs 17171 LSpanclspn 20844 HLchlt 38759 LHypclh 39394 DVecHcdvh 40488 DIsoHcdih 40638 ocHcoch 40757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-riotaBAD 38362 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-n0 12495 df-z 12581 df-uz 12845 df-fz 13509 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-0g 17414 df-proset 18278 df-poset 18296 df-plt 18313 df-lub 18329 df-glb 18330 df-join 18331 df-meet 18332 df-p0 18408 df-p1 18409 df-lat 18415 df-clat 18482 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-subg 19069 df-cntz 19259 df-lsm 19582 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-drng 20615 df-lmod 20734 df-lss 20805 df-lsp 20845 df-lvec 20977 df-lsatoms 38385 df-oposet 38585 df-ol 38587 df-oml 38588 df-covers 38675 df-ats 38676 df-atl 38707 df-cvlat 38731 df-hlat 38760 df-llines 38908 df-lplanes 38909 df-lvols 38910 df-lines 38911 df-psubsp 38913 df-pmap 38914 df-padd 39206 df-lhyp 39398 df-laut 39399 df-ldil 39514 df-ltrn 39515 df-trl 39569 df-tendo 40165 df-edring 40167 df-disoa 40439 df-dvech 40489 df-dib 40549 df-dic 40583 df-dih 40639 df-doch 40758 |
This theorem is referenced by: dochsnnz 40860 lcfl8b 40914 lclkrlem2c 40919 lcfrlem23 40975 lcfrlem26 40978 lcfrlem36 40988 mapdval4N 41042 mapdsn 41051 hdmapglem7a 41337 |
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